The Republican pair also said that Marine Corps General James Mattis was an "impressive" prospect for defence secretary.

Vice president-elect Mr Pence, who is leading the search for Mr Trump's cabinet members, told CBS on Sunday: " Governor Romney is under active and serious consideration to serve as secretary of state of the United States."

The remark made clear Mr Trump and Mr Romney's meeting a day earlier in New Jersey was more than a public attempt to bury the hatchet after trading insults during the campaign.

On Sunday, the billionaire interrupted his tweeted criticisms of Saturday Night Live, the hit musical Hamilton and Democrats to state "General James 'Mad Dog' Mattis, who is being considered for secretary of defence, was very impressive yesterday. A true General's General!"

The comments were indications Mr Trump could be nearing additional Cabinet announcements soon as he works toward rounding out his foreign policy and national security teams.

He took a hard line on both subjects during the presidential campaign, proposing a temporary ban on Muslims entering the country and more.

Even as Mr Trump and his team discussed some of the weightiest matters facing the country, the president-elect continued tweeting about an array of matters.

He also complained that Saturday Night Live, which thrives on making fun of politicians, is "biased" and not funny. The night before, actor Alec Baldwin portrayed Mr Trump as Googling "What is Isis?"

Mr Trump also insisted again the cast and producers of Hamilton should apologise after the lead actor addressed Mr Pence from the stage on Friday night, telling the vice president-elect that "diverse America" was "alarmed and anxious" about the incoming administration.

Mr Pence told CBS that he enjoyed the show and was not offended.

Meanwhile, Mr Trump received more visitors to his golf club in New Jersey on Sunday.

His schedule in Bedminster was to include former New York City Mayor Rudy Giuliani and Kansas Secretary of State Kris Kobach.

Mr Trump said he may announce some of his picks on Sunday.

The president-elect started filling key administration positions on Friday, picking Alabama Senator Jeff Sessions for attorney general and Kansas Representative Mike Pompeo to head the CIA, signalling a sharp rightward shift in US security policy as he begins to form his Cabinet.

Mr Trump also named retired Lieutenant General Michael Flynn as his national security adviser.